A Bayes method for a monotone hazard rate via S-paths

Annals of Statistics

Volumn 34, Issue 2, 2006, Pages 820-836

  Ho, Man Wai ^a  

^a NATIONAL UNIVERSITY OF SINGAPORE   (Singapore)

Author keywords

Completely random measure; Gibbs sampler; Markov chain Monte Carlo; Proportional hazard model; Random partition; Rao Blackwellization; Weighted gamma process

Indexed keywords

EID: 33746253028     PISSN: 00905364     EISSN: None     Source Type: Journal    
DOI: 10.1214/009053606000000047     Document Type: Article

References (50)

1. AALEN, O. (1975). Statistical inference for a family of counting processes. Ph.D. dissertation, Univ. California, Berkeley.
2. AALEN, O. (1978). Nonparametric inference for a family of counting processes. Ann. Statist. 6 701-726. MR0491547
3. ANDERSEN, P. K., BORGAN, O., GILL, R. D. and KEIDING, N. (1993). Statistical Models Based on Counting Processes. Springer, New York. MR1198884
4. BARLOW, R. E., BARTHOLOMEW, D. J., BREMNER, J. M. and BRUNK, H. D. (1972). Statistical Inference Under Order Restrictions. The Theory and Application of Isotonic Regression. Wiley, New York. MR0326887
5. BARNDORFF-NIELSEN, O. E. and SHEPHARD, N. (2001). Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics (with discussion). J. R. Stat. Soc. Ser. B Stat. Methodol. 63 167-241. MR1841412
6. BARRON, A., SCHERVISH, M. J. and WASSERMAN, L. (1999). The consistency of posterior distributions in nonparametric problems. Ann. Statist. 27 536-561. MR1714718
7. BLACKWELL, D, and MACQUEEN, J. B. (1973). Ferguson distributions via Pólya urn schemes. Ann. Statist. 1 353-355. MR0362614
8. BRIX, A. (1999). Generalized gamma measures and shot-noise Cox processes. Adv. in Appl. Probab. 31 929-953. MR1747450
9. BRUNNER, L. J. (1995). Bayesian linear regression with error terms that have symmetric unimodal densities. J. Nonparametr. Statist. 4 335-348. MR1366780
10. BRUNNER, L. J. and Lo, A. Y. (1989). Bayes methods for a symmetric unimodal density and its mode. Ann. Statist. 17 1550-1566. MR1026299
11. BRUNNER, L. J. and Lo, A. Y. (1994). Nonparametric Bayes methods for directional data. Canad. J. Statist. 22401-412. MR1309323
12. Cox, D. R. (1972). Regression models and life-tables (with discussion). J. Roy. Statist. Soc. Ser. B 34 187-220. MR0341758
13. DRǍGICHI, L. and RAMAMOORTHI, R. V. (2003). Consistency of Dykstra-Laud priors. Sankhya 65 464-481. MR2028910
14. DYKSTRA, R. L. and LAUD, P. (1981). A Bayesian nonparametric approach to reliability. Ann. Statist. 9 356-367. MR0606619
15. FELLER, W. (1968). An Introduction to Probability Theory and Its Applications 1, 3rd ed. Wiley, New York. MR0228020
16. GEMAN, S. and GEMAN, D. (1984). Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Trans. Pattern Analysis Machine Intelligence 6 721-741.
17. GRENANDER, U. (1956). On the theory of mortality measurement. II. Skand. Aktuarietidskr. 39 125-153. MR0093415
18. HALD, A. (1981). T. N. Thiele's contributions to statistics. Internat. Statist. Rev. 49 1-20. MR0623007
19. HALL, P., HUANG, L.-S., GIFFORD, J. A. and GIJBELS, I. (2001). Nonparametric estimation of hazard rate under the constraint of monotonicity. J. Comput. Graph. Statist. 10 592-614. MR1939041
20. HASTINGS, W. K. (1970). Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57 97-109.
21. HAYAKAWA, Y., ZUKERMAN, J., PAUL, S. and VIGNAUX, T. (2001). Bayesian nonparametric testing of constant versus nondecreasing hazard rates. In System and Bayesian Reliability: Essays in Honor of Professor Richard E. Barlow on His 10th Birthday ( Y. Hayakawa, T. Irony and M. Xie, eds.) 391-406. World Scientific, River Edge, NJ. MR1896794
22. Ho, M.-W. (2002). Bayesian inference for models with monotone densities and hazard rates. Ph.D. dissertation, Dept. Information and Systems Management, Hong Kong Univ. of Science and Technology.
23. Ho, M.-W. (2005). A Bayes method for an asymmetric unimodal density with mode at zero. Unpublished manuscript.
24. Ho, M.-W. (2006). Bayes estimation of a symmetric unimodal density via S-paths. J. Comput. Graph. Statist. To appear.
25. Ho, M.-W. and Lo, A. Y. (2001). Bayesian nonparametric estimation of a monotone hazard rate. In System and Bayesian Reliability: Essays in Honor of Professor Richard E. Barlow on His 70th Birthday (Y. Hayakawa, T. Irony and M. Xie, eds.) 301-314. World Scientific, River Edge, NJ. MR1896789
26. HUANG, J. and WELLNER, J. A. (1995). Estimation of a monotone density or monotone hazard under random censoring. Scand. J. Statist. 22 3-33. MR1334065
27. IBRAHIM, J. G., CHEN, M.-H. and MACEACHERN, S. N. (1999). Bayesian variable selection for proportional hazards models. Canad. J. Statist. 27 701-717. MR1767142
28. ISHWARAN, H. and JAMES, L. F. (2001). Gibbs sampling methods for stick-breaking priors. J. Amer. Statist. Assoc. 96 161-173. MR1952729
29. ISHWARAN, H. and JAMES, L. F. (2003). Generalized weighted Chinese restaurant processes for species sampling mixture models. Statist. Sinica 13 1211-1235. MR2026070
30. ISHWARAN, H. and JAMES, L. F. (2004). Computational methods for multiplicative intensity models using weighted gamma processes: Proportional hazards, marked point processes and panel count data. J. Amer. Statist. Assoc. 99 175-190. MR2054297
31. JAMES, L. F. (2003). Bayesian calculus for gamma processes with applications to semiparametric intensity models. Sankhyä 65 179-206. MR2016784
32. JAMES, L. F. (2005). Bayesian Poisson process partition calculus with an application to Bayesian Levy moving averages. Ann. Statist. 33 1771-1799. MR2166562
33. JAMES, L. F. and LAU, J. W. (2005). A class of generalized hyperbolic continuous time integrated stochastic volatility likelihood models. Available at www.arXiv.org/abs/math.ST/ 0503056.
34. JAMES, L. F., LIJOI, A. and PRÜNSTER, I. (2005). Bayesian inference via classes normalized random measures. Available at www.arXiv.org/abs/math.ST/ 0503394.
35. KALBFLEISCH, J. D. (1978). Non-parametric Bayesian analysis of survival time data. J. Roy. Statist. Soc. Ser. B 40 214-221. MR0517442
36. KINGMAN, J. F. C. (1967). Completely random measures. Pacific J. Math. 21 59-78. MR0210185
37. KINGMAN, J. F. C. (1993). Poisson Processes. Oxford Univ. Press, New York. MR1207584
38. LO, A. Y. (1982). Bayesian nonparametric statistical inference for Poisson point processes. Z. Wahrsch. Verw. Gebiete 59 55-66. MR0643788
39. LO, A. Y. (1984). On a class of Bayesian nonparametric estimates, I. Density estimates. Ann. Statist. 12 351-357. MR0733519
40. LO, A. Y., BRUNNER, L. J. and CHAN, A. T. (1996). Weighted Chinese restaurant processes and Bayesian mixture models. Research report, Hong Kong Univ. of Science and Technology. Available at www.erin.utoronto.ca/~jbrunner/ papers/wcr96.pdf.
41. LO, A. Y. and WENG, C.-S. (1989). On a class of Bayesian nonparametric estimates. II. Hazard rate estimates. Ann. Inst. Statist. Math. 41 227-245. MR1006487
42. LO, S. H. and PADIA, E. (1992), On estimation of a survival function in reliability theory based on censored data. Preprint, Columbia Univ.
43. MCCULLAGH, P. (1987). Tensor Methods in Statistics. Chapman and Hall, London. MR0907286
44. MEYN, S. P. and TWEEDIE, R. L. (1993). Markov Chains and Stochastic Stability. Springer, Berlin. MR1287609
45. MYKYTYN, S. W. and SANTNER, T. J. (1981). Maximum likelihood estimation of the survival function based on censored data under hazard rate assumptions, Comm. Statist. A - Theory Methods 10 1369-1387. MR0625034
46. PADGETT, W. J. and WEI, L. J. (1980). Maximum likelihood estimation of a distribution function with increasing failure rate based on censored observations. Biometrika 67 470-474. MR0581744
47. PRAKASA RAO, B. L. S. (1970), Estimation of distributions with monotone failure rate. Ann. Math. Statist. 41 507-519. MR0260133
48. TIERNEY, L. (1994). Markov chains for exploring posterior distributions (with discussion). Ann. Statist. 22 1701-1762. MR1329166
49. VILLALOBOS, M. and WAHBA, G. (1987). Inequality-constrained multivariate smoothing splines with application to the estimation of posterior probabilities. J. Amer. Statist. Assoc. 82 239-248. MR0883352
50. WOLPERT, R. L. and ICKSTADT, K. (1998). Poisson/gamma random field models for spatial statistics. Biometrika 85 251-267. MR1649114